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Algebraic Expressions and Identities - Addition and Subtraction of Algebraic Expressions

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An Algebraic Expression is formed from variables and constants using operations like addition and subtraction. For example, in the expression 4x2−3xy4x^2 - 3xy, 4x24x^2 and −3xy-3xy are the terms. Think of terms as individual 'building blocks' separated by plus or minus signs.

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Terms are made of factors; for instance, 5xy5xy has factors 5,x, and y5, x, \text{ and } y. The numerical factor is called the numerical coefficient. In the term −7ab-7ab, the coefficient is −7-7. Visually, you can imagine a term as a product box containing a number and several variables.

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Like Terms are terms that have the same algebraic factors (identical variables with identical exponents). For example, 2x2y2x^2y and −5x2y-5x^2y are like terms. Unlike Terms have different algebraic factors, such as 3x3x and 3y3y. Like terms are like 'matching socks' that can be paired together, while unlike terms are different items that must remain separate.

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Expressions are classified by the number of terms: a Monomial has one term (5x5x), a Binomial has two terms (a+ba + b), and a Trinomial has three terms (x2+x+1x^2 + x + 1). Any expression with one or more terms is called a Polynomial. You can visualize this as a train where each 'car' is a term.

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To add algebraic expressions, we group like terms together and find the sum of their numerical coefficients. For example, 3x+4x=(3+4)x=7x3x + 4x = (3+4)x = 7x. Imagine combining two piles of identical coins; the total value is simply the sum of the count of coins.

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The Column Method for addition involves writing expressions in separate rows, such that like terms are aligned vertically in the same column. It looks similar to standard multi-digit addition where units are under units and tens are under tens. If a term is missing in one expression, we leave a blank space or use 00 as a placeholder.

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Subtraction of expressions is performed by adding the 'Additive Inverse' of the expression to be subtracted. This means you change the sign of every term in the expression being subtracted (change ++ to −- and −- to ++). Visually, this is like flipping the direction of every arrow in a vector diagram before combining them.

📐Formulae

Sum of Like Terms=(Sum of Numerical Coefficients)×Common Algebraic Factors\text{Sum of Like Terms} = (\text{Sum of Numerical Coefficients}) \times \text{Common Algebraic Factors}

Subtraction Rule: A−B=A+(−B)\text{Subtraction Rule: } A - B = A + (-B)

Identity: p(x)+q(x)=Combining all like terms from both polynomials\text{Identity: } p(x) + q(x) = \text{Combining all like terms from both polynomials}

(axn)+(bxn)=(a+b)xn(ax^n) + (bx^n) = (a + b)x^n

(axn)−(bxn)=(a−b)xn(ax^n) - (bx^n) = (a - b)x^n

💡Examples

Problem 1:

Add the expressions: 7x2−4x+57x^2 - 4x + 5 and 9x−109x - 10.

Solution:

Step 1: Write the expressions together: (7x2−4x+5)+(9x−10)(7x^2 - 4x + 5) + (9x - 10) \ Step 2: Group like terms: 7x2+(−4x+9x)+(5−10)7x^2 + (-4x + 9x) + (5 - 10) \ Step 3: Simplify coefficients: 7x2+(5)x+(−5)7x^2 + (5)x + (-5) \ Step 4: Final Result: 7x2+5x−57x^2 + 5x - 5

Explanation:

We identify that 7x27x^2 has no like term, −4x-4x and 9x9x are like terms, and 55 and −10-10 are constants (like terms). We sum the coefficients of the matching variables.

Problem 2:

Subtract 3a2−4b2+5ab−33a^2 - 4b^2 + 5ab - 3 from 7a2−2b2+3ab−107a^2 - 2b^2 + 3ab - 10.

Solution:

Step 1: Set up the subtraction: (7a2−2b2+3ab−10)−(3a2−4b2+5ab−3)(7a^2 - 2b^2 + 3ab - 10) - (3a^2 - 4b^2 + 5ab - 3) \ Step 2: Change the signs of the second expression: 7a2−2b2+3ab−10−3a2+4b2−5ab+37a^2 - 2b^2 + 3ab - 10 - 3a^2 + 4b^2 - 5ab + 3 \ Step 3: Rearrange like terms: (7a2−3a2)+(−2b2+4b2)+(3ab−5ab)+(−10+3)(7a^2 - 3a^2) + (-2b^2 + 4b^2) + (3ab - 5ab) + (-10 + 3) \ Step 4: Simplify: 4a2+2b2−2ab−74a^2 + 2b^2 - 2ab - 7

Explanation:

In subtraction, the most important step is distributing the negative sign to every term in the subtrahend. Then, we proceed with the standard addition of like terms.